The joint probability distribution of thr random variables X and Y is given below: f(x,y)={(cxy , 0<x<2,0<y<x),(0,, \text{ other }):}

Khadija Wells 2021-09-06 Answered

The joint probability distribution of thr random variables X and Y is given below:
f(x,y)={cxy0<x<2,0<y<x0 other 
a.Find the value of the constant.
b.Calculate the covariance and the correlation of the X and Y random variables.
c. Calculate the expected value of the random variable Z=2X3Y+2

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Expert Answer

Theodore Schwartz
Answered 2021-09-07 Author has 99 answers

Step 1
Given:
f(x,y)={cxy0<x<2,0<y<x0 other 
a. value of constant c,
020xcxydydx=1
02cx[y22]0xdx=1
c202x×x2dx=1
c2[x44]02=1
2c=1
c=12
Step 2
b.
we have given a joint pdf,
f(x,y)={cxy0<x<2,0<y<x0 other 
marginal pdf of x,
f1(x)=0xf(x,y)dy
f1(x)=0x12xydy
f1(x)=12x×(y22)0x
f1(x)={14x30<x<20 otherwise 
E(x)=02x×f1(x)dx
E(x)=02×x34dx
E(x)=14(x55)02
E(x)=85
Step 3
E(x2)=02x2×f1(x)dx
E(x2)=02x2×x

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